The current $I(t)$ at time $t$ in an electrical circuit subject to an applied voltage $V(t)$ obeys the equation

$$L \frac{d^{2} I}{d t^{2}}+R \frac{d I}{d t}+\frac{1}{C} I=\frac{d V}{d t}$$

where $R, L$ and $C$ are the constant resistance, inductance and capacitance of the circuit with $R \geqslant 0, L>0$ and $C>0$.

(a) In the case $R=0$ and $V(t)=0$, show that there exist time-periodic solutions of frequency $\omega_{0}$, which you should find.

(b) In the case $V(t)=H(t)$, the Heaviside function, calculate, subject to the condition

$$R^{2}>\frac{4 L}{C}$$

the current for $t \geqslant 0$, assuming it is zero for $t<0$.

(c) If $R>0$ and $V(t)=\sin \omega_{0} t$, where $\omega_{0}$ is as in part (a), show that there is a timeperiodic solution $I_{0}(t)$ of period $T=2 \pi / \omega_{0}$ and calculate its maximum value $I_{M}$.

(i) Calculate the energy dissipated in each period, i.e., the quantity

$$D=\int_{0}^{T} R I_{0}(t)^{2} d t$$

Show that the quantity defined by

$$Q=\frac{2 \pi}{D} \times \frac{L I_{M}^{2}}{2}$$

satisfies $Q \omega_{0} R C=1$.

(ii) Write down explicitly the general solution $I(t)$ for all $R>0$, and discuss the relevance of $I_{0}(t)$ to the large time behaviour of $I(t)$.